# Getting Started¶

## Installation¶

If you use pip, just type

pip install pyqubo


You can install from the source code like

git clone https://github.com/recruit-communications/pyqubo.git
cd pyqubo
python setup.py install


## QUBO and Ising Model¶

If you want to solve a combinatorial optimization problem by quantum or classical annealing machines, you need to represent your problem in QUBO (Quadratic Unconstrained Binary Optimization) or Ising model. PyQUBO converts your problem into QUBO or Ising model format.

The objective function of QUBO is defined as:

$\sum_{i \le j}q_{ij}x_{i}x_{j}$

where $$x_{i}$$ represents a binary variable which takes 0 or 1, and $$q_{ij}$$ represents a quadratic coefficient. Note that $$q_{ii}x_{i}x_{i}=q_{ii}x_{i}$$, since $$x_{i}^2=x_{i}$$. Thus, the above expression includes linear terms of $$x_{i}$$.

The objective function of Ising model is defined as:

$\sum_{i}h_{i}s_{i} + \sum_{i<j}J_{ij}s_{i}s_{j}$

where $$s_{i}$$ represents spin variable which takes -1 or 1, $$h_{i}$$ represents an external magnetic field and $$J_{ij}$$ represents an interaction between spin $$i$$ and $$j$$.

## Basic Usage¶

With PyQUBO, you can construct QUBOs with 3 steps:

1. Define the Hamiltonian.
>>> from pyqubo import Spin
>>> s1, s2, s3, s4 = Spin("s1"), Spin("s2"), Spin("s3"), Spin("s4")
>>> H = (4*s1 + 2*s2 + 7*s3 + s4)**2

1. Compile the Hamiltonian to get a model.
>>> model = H.compile()

1. Call ‘to_qubo()’ to get QUBO coefficients.
>>> qubo, offset = model.to_qubo()
>>> pprint(qubo) # doctest: +SKIP
{('s1', 's1'): -160.0,
('s1', 's2'): 64.0,
('s1', 's3'): 224.0,
('s1', 's4'): 32.0,
('s2', 's2'): -96.0,
('s2', 's3'): 112.0,
('s2', 's4'): 16.0,
('s3', 's3'): -196.0,
('s3', 's4'): 56.0,
('s4', 's4'): -52.0}
>>> print(offset)
196.0


In this example, you want to solve Number Partitioning Problem with a set S = {4, 2, 7, 1}. The hamiltonian $$H$$ is represented as

$H = (4 s_{1} + 2 s_{2} + 7 s_{3} + s_{4})^2$

where $$s_{i}$$ is a $$i$$ th spin variable which indicates a group the $$i$$ th number should belong to. In PyQUBO, spin variables are internally converted to binary variables via the relationship $$x_{i} = (s_{i}+1)/2$$. The QUBO coefficents and the offset returned from Model.to_qubo() represents the following objective function:

$\begin{split}&-160 x_{1}x_{1} + 64 x_{1}x_{2} + 224 x_{1}x_{3} + 32 x_{1}x_{4} - 96 x_{2}x_{2}\\ &+ 112 x_{2}x_{3} + 16 x_{2}x_{4} - 196 x_{3}x_{3} + 56 x_{3}x_{4} - 52 x_{4}x_{4} + 196\end{split}$
1. Call ‘to_ising()’ to get Ising coefficients.

If you want to get the coefficient of the Ising model, just call to_ising() method like below.

>>> linear, quadratic, offset = model.to_ising()
>>> pprint(linear) # doctest: +SKIP
{'s1': 0.0, 's2': 0.0, 's3': 0.0, 's4': 0.0}
{('s1', 's2'): 16.0,
('s1', 's3'): 56.0,
('s1', 's4'): 8.0,
('s2', 's3'): 28.0,
('s2', 's4'): 4.0,
('s3', 's4'): 14.0}
>>> print(offset)
70.0


where linear represents external magnetic fields $$h$$, quadratic represents interactions $$J$$ and offset represents the constant value in the objective function below.

$16 s_{1}s_{2} + 56 s_{1}s_{3} + 8 s_{1}s_{4} + 28 s_{2}s_{3} + 4 s_{2}s_{4} + 14 s_{3}s_{4} + 70$

## Variable: Binary and Spin¶

When you define a Hamiltonian, you can use Binary or Spin class to represent $$\{0,1\}$$ or $$\{1,-1\}$$ variable.

Example: If you want to define a Hamiltonian with binary variables $$x \in \{0, 1\}$$, use Binary.

>>> from pyqubo import Binary
>>> x1, x2 = Binary('x1'), Binary('x2')
>>> H = 2*x1*x2 + 3*x1
>>> pprint(H.compile().to_qubo()) # doctest: +SKIP
({('x1', 'x1'): 3.0, ('x1', 'x2'): 2.0, ('x2', 'x2'): 0.0}, 0.0)


Example: If you want to define a Hamiltonian with spin variables $$s \in \{-1, 1\}$$, use Spin.

>>> from pyqubo import Spin
>>> s1, s2 = Spin('s1'), Spin('s2')
>>> H = 2*s1*s2 + 3*s1
>>> pprint(H.compile().to_qubo()) # doctest: +SKIP
({('s1', 's1'): 2.0, ('s1', 's2'): 8.0, ('s2', 's2'): -4.0}, -1.0)


## Solve QUBO by dimod Sampler¶

PyQUBO model can output the BinaryQuadraticModel(BQM). You can solve BQM by using Sampler class. Sampler is an abstract class defined by dimod package. Various kinds of sampler class, such as SimulatedAnnealingSampler or DWaveSampler, inherits Sampler class.

First, we craete BQM object using to_bqm() method. (If you want to use DWaveSampler which only takes integer-indexed QUBO, you can simply do like to_bqm(index_label=True).)

>>> from pyqubo import Binary
>>> x1, x2 = Binary('x1'), Binary('x2')
>>> H = (x1 + x2 - 1)**2
>>> model = H.compile()
>>> bqm = model.to_bqm()


Next, we create neal.SimulatedAnnealingSampler and use sample() method to get the solutions of QUBO as SampleSet. You can use Model.decode_sampleset() to interpret the sampleset object, and it returns decoded_samples which is a list of pyqubo.DecodedSample object.

>>> import neal
>>> sa = neal.SimulatedAnnealingSampler()
>>> decoded_samples = model.decode_sampleset(sampleset)
>>> best_sample = min(decoded_samples, key=lambda x: x.energy)
>>> pprint(best_sample.sample)
{'x1': 0, 'x2': 1}


## Array of Variables¶

Array class represents a multi-dimensional array of Binary or Spin.

Example: You can access each element of the matrix with an index like:

>>> from pyqubo import Array
>>> x = Array.create('x', shape=(2, 3), vartype='BINARY')
>>> x[0, 1] + x[1, 2]
(Binary(x[0][1])+Binary(x[1][2]))


Example: You can use Array to represent multiple spins in the example of partitioning problem above.

>>> from pyqubo import Array
>>> numbers = [4, 2, 7, 1]
>>> s = Array.create('s', shape=4, vartype='SPIN')
>>> H = sum(n * s for s, n in zip(s, numbers))**2
>>> model = H.compile()
>>> qubo, offset = model.to_qubo()
>>> pprint(qubo) # doctest: +SKIP
{('s[0]', 's[0]'): -160.0,
('s[0]', 's[1]'): 64.0,
('s[0]', 's[2]'): 224.0,
('s[0]', 's[3]'): 32.0,
('s[1]', 's[1]'): -96.0,
('s[1]', 's[2]'): 112.0,
('s[1]', 's[3]'): 16.0,
('s[2]', 's[2]'): -196.0,
('s[2]', 's[3]'): 56.0,
('s[3]', 's[3]'): -52.0}


## Placeholder¶

If you have a parameter that you will probably update, such as the strength of the constraints in your hamiltonian, using Placeholder will save your time. If you define the parameter by Placeholder, you can specify the value of the parameter after compile. This means that you don’t have to compile repeatedly for getting QUBOs with various parameter values. It takes longer time to execute a compile when the problem size is bigger. In that case, you can save your time by using Placeholder.

Example: If you have an objective function $$2a+b$$, and a constraint $$a+b=1$$ whose hamiltonian is $$(a+b-1)^2$$ where $$a,b$$ is qbit variable, you need to find the penalty strength $$M$$ such that the constraint is satisfied. Thus, you need to create QUBO with different values of $$M$$. In this example, we create QUBO with $$M=5.0$$ and $$M=6.0$$.

In the first code, we don’t use placeholder. In this case, you need to compile the hamiltonian twice to get a QUBO with $$M=5.0$$ and $$M=6.0$$.

>>> from pyqubo import Binary
>>> a, b = Binary('a'), Binary('b')
>>> M = 5.0
>>> H = 2*a + b + M*(a+b-1)**2
>>> model = H.compile()
>>> qubo, offset = model.to_qubo() # QUBO with M=5.0
>>> M = 6.0
>>> H = 2*a + b + M*(a+b-1)**2
>>> model = H.compile()
>>> qubo, offset = model.to_qubo() # QUBO with M=6.0


If you don’t want to compile twice, define $$M$$ by Placeholder.

>>> from pyqubo import Placeholder
>>> a, b = Binary('a'), Binary('b')
>>> M = Placeholder('M')
>>> H = 2*a + b + M*(a+b-1)**2
>>> model = H.compile()
>>> qubo, offset = model.to_qubo(feed_dict={'M': 5.0})


You get a QUBO with different value of M without compile

>>> qubo, offset = model.to_qubo(feed_dict={'M': 6.0})


The actual value of the placeholder $$M$$ is specified in calling Model.to_qubo() as a value of the feed_dict.

## Validation of Constraints¶

When you get a solution from the Sampler, Model.decode_sample() decodes the sample and returns DecodedSample object.

Example: You are solving a partitioning problem.

>>> from pyqubo import Binary, Constraint
>>> a, b = Binary('a'), Binary('b')
>>> M = 5.0 # strength of the constraint
>>> H = 2*a + b + M * Constraint((a+b-1)**2, label='a+b=1')
>>> model = H.compile()


Let’s assume that you get a solution {'a': 0, 'b': 1} from the solver.

>>> raw_solution = {'a': 0, 'b': 1} # solution from the solver
>>> decoded_sample = model.decode_sample(raw_solution, vartype='BINARY')
>>> pprint(decoded_sample.sample)
{'a': 0, 'b': 1}
>>> pprint(decoded_sample.constraints())
{'a+b=1': (True, 0.0)}
>>> pprint(decoded_sample.constraints(only_broken=True))
{}


You can access to the dict of the sample via decoded_sample.sample. You can also access to the value of the constraint of the Hamiltonian via decoded_sample.constraints(). If you specify the argument only_broken=True, only broken constraint will be returned. If the empty dict is returned, it indicates that there is no broken constraint corresponding to the given sample.